• History of numbers
• Base 10
• History of Hindu-Arabic Numbers
• Other base systems
• Maths in base 12

• Gain an awareness of the history of numbers
• State the difference between positional and non-positional number systems, and the confusion which can arise in non-positional systems
• Express numbers in any base in a convenient format
• Perform calculations with numbers in other bases

• Likely began as a way of counting items in the environment.
• Humans came up with words and symbols to answer “how many?” or “how much?”

Lebombo bone

• 44,200-43,000 years old
• found in South Africa/Mozambique.
• Earliest example of a counting stick (i.e. not purely decorative)

• The most commonly used system for representing numbers
• Positional system: the position of a number in the sequence represents its magnitude.
• Indian mathematicians used the same symbol in different places to represent higher numbers, rather than adding more symbols.
• Chinese, Mayan, and Babylonians (among others) developed a positional system independently

• We are used to counting using Hindu-Arabic numbers:

  \(1, 2, 3, …, 9, 10, 11, 12, …, 103, 2476, \ldots\)

• Numbers can be represented using a positional table (based on Vederic logic).

• Each row represents one digit (one symbol)
• Once we run out of symbols, we add another row…

• …the second row counts up in multiples of 10.

• This is how numbers work in some languages…

  E.g., Thai:

  • https://www.facebook.com/watch/?v=1976533773114013
  • https://youtu.be/eglcvo5UUyQ

For example, the number 14 can be expressed as:

• Wrote with a tablet and stylus
• Their numbers were made by stamping the tablet with the stylus

• Hebrew letters and numbers use the same symbols

• In standard notation 18 is XVIII (10, 5, and 3 1s)
• but the Roman 18th Legion (of Teuton Forest fame) wrote their number as XIIX!
• And is 499:
  • CDXCIX (CD = 500 - 100, XC = 100 - 10, IX = 10 - 1)
  • LDVLIV (LD = 500 - 50, VL = 50 - 5, IV = 5 - 1)
  • XDIX (XD = 500 - 10, IX = 10 - 1) (10 before 50, 1 before 10)
  • VDIV (VD = 500 - 5, IV = 5 - 1)
  • ID (500 - 1)

• There are as many bases as there are symbols.

• We will express them using the same positional table format.

• But first…

• Originate from the Vedric period of Indian history (1500–500 BCE)

• [Taittiriya Samhita] used words to represent powers of 10

  (e.g. dasa = 101, shatha = 102, sahasra = 103)

• The big innovation was to include 0 as a number, not just a placeholder.

• Accredited to Brahmagupta (c 598CE - c 668CE) though evidence points to it being used in the Bakhshali manuscript (224–383 CE or 885–993 CE)

The hindu-arabic numbers

• Base 10, positional notation in powers of 10, and 0.

• Adopted by Abbasid Caliphate under Caliph al-Mansur through sponsorship of the translation movement.

• Move to translate secular books from Greek, Persian, Syriac, and many other languages into Arabic

• Formed part of the “Islamic Golden Age”

• Al-Khwārizmī and Al-Kindi were strong proponents of this system

• Al-Khwārizmī (c 780 - c 847)
  • Persian mathematician and polymath
  • Book Kitāb al-ḥisāb al-hindī (‘Book of Indian computation’) used this system
• Al-Kindi (c 801 - c 873)
  • known as the “father of Arabic philosophy”

–

Map showing the Abbhasid Kaliphate

• Appeared in Europe in the Codex Vigilanus (976 CE)

• Promoted by mathematicians such as Fibonacci (c 1170 – c 1245)

• Became more popular after the invention of the printing press (post 1482 CE)

  • Made the process of printing easier
  • Conjectured that this is also why \(x\) was chosen for the unknown variable

“It is India that gave us the ingenious method of expressing all numbers by the means of ten symbols, each1 symbol receiving a value of position, as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit, but its very simplicity, the great ease which it has lent to all computations, puts our arithmetic in the first rank of useful inventions, and we shall appreciate the grandeur of this achievement when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest minds produced by antiquity.”

– Pierre-Simon Laplace (1749–1827)

0 “is big. You just won’t believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it’s a long way down the road to the chemist’s, but that’s just peanuts” compared to 0 (Douglas Adams, The Hitchhiker’s Guide to the Galaxy talking about space)

• A revolutionary idea, relying on a philosophy which permits the idea of nothingness

• Modern mathematics would not be possible without zero

• Neither would mathematics with computers.

Other than base 10, which other bases are used in daily life?

• 60 seconds in a minute
• 60 minutes in an hour (often counted in multiples of 5)
• 24 hours in a day (or two groups of 12)
• 7 days in a week
• 28,29,30,31 days in a month
• 4 + d days for d from 0 to 3 weeks in a month
• 12 months in a year
• 52 + d days for d = 1 or d = 2, weeks in a year

• Base 10 (humans love base 10 so much that some early computers were in base 10)

• Base 2 (Binary): how processors work, how data is stored

• Base 8: common way of representing a byte (8 bits)

• Base 16: another common representation for binary encoded information, e.g., colours

• Base 12 – Ed’s favourite… let’s see if he can convince us!

Convert the number \(13057\) from base \(8\) into base \(10\).

• To encode base 12, we need two additional symbols
• We can’t use base 10 symbols 10 and 11
• Is 111 in base 12 equal to
  • \(1 \cdot 144 + 1 \cdot 12 + 1 \cdot 1\)
  • \(1 \cdot 12 + 11 \cdot 1\)
  • \(11 \cdot 12 + 1 \cdot 1\)
• We will let
  • \(10\) in base 10 be A
  • and \(11\) be B

• Likewise, we need 6 extra symbols for base 16:

  0 1 2 3 4 5 6 7 8 9 A B C D E F

• In general, we’ll use capital letters for additional symbols, until we run out of those…